4 J 1 bmd . unz b 1 rql~8n~7t:uu(;'1~yii tilun?~ tmu~auq~i la t : kui .,b B hmldlnoatun'~ a iinfasid?nanou uszlunauhuun t vii&dnr'.raauif~d48a

v 4 w d ' j W-1 mode ~ut:uu%uqiiaq h u ~ d ~ ~ ~ t : i / ~ ~ ~ u ~ t ~ u u ~IIII?UUO~R,PI ~ u n l un71

4 4 4" I 6 1 I I 8nnqr:uuwLqr t nnounnmur ~n~uuann~zn'7noayn~n t n eun uuq~iinndnulmn?l~

et.r Damped Driven One-dimnsioml Harmonic Oscillator

I I I I I 4 4 h o u i ~ t a u Ld3 ln n ~ r u n ~ . r s u ~ ~ ~ ~ r z a e ~ ~ n ~ r r : n n a ~ n a ~ n i ~ a ~ ~ u n n a n k ~ ~ u \ ~ v 4 J

i g i au\d n i t i 8nn~imiun?uaarn?t inaauwnaiumqr;qdut Jgu'ndun ~:MSI&I t :uu 4 v ~d~iln'iudtniurnin 19 i~ i louf iu t ~dn~mfidlanauiuiun t.zn'71ui~nr Ja iuf imwi ;J

I I

constant per unit mass ufal ?unJ7~7q? dampin8 constant h t t t q ~ n ~ l t ! M

aafi3a i nnlun'finq~ x n i t r f n r i n a u g r ~ ~ a ~ i i ; 8 x(t) ~ n u d n h ~ a r z h r c k n L 4 ' J L.41 '

-bbtr(t) L tue~uirinfidf ~ r a ~ n i n ~ ~ a d i ~ K = %t uunan~luilur ~n~uuana&c 4 4 1 ' dus~a~zoadh t anuuua~;~dnnaan\d i r a u ~ n ~ u n ~ i r n i P J ~ U u,, i ra \ r4~ i twd i i t J

z flJJnltul5~ inhomogeneous second-order linear differential equation

88

1 I ~ u i u u r m I .r~:~wrtumtoin'~ P!Y L uau'q~~u"iis qnluuon L ?unil Transient decay .f 4

free oscillations wn~unliqaqnv 1 n t u h u m n l l (m. 9 ) t j a ~ u t j u

?i r :uuutun?r oofie L anun:6ud~nd~mna 4 i iqn;?nri~ crit ically daopad .I v I 4 I wn$ fiwnrl;t: ~ffuln';? nrtiiLi!o v un~uaun71 ro ~ 1 ~ n ~ o o m 3 0 ~ n n 1 !un7?

I

under damped i ~ u n r $ ~ i d u . [ ~ u i n n ~ ~ ~ V I ~ I : i iIunrddi;n?rooda iaf~if loufi~? l f l ~ I I I

UR: L ~ I ~ I ~ T : ~ T I ~ ~ J I ? W ~ U t ~ u ~ o n d i ~ u u i i u a e I 4 -(Wrt ji?n4d,:H~~4Hu4rou.[n~ v

ao~niroonria~ lm k&mitr i i~aa~al(ninQnn?~unn~un?ldr:uiurno.r~~ayriu;fi n a ?

qa4munlr (n. b) I~~U&I;I e -(Wrt ~ ~ I R P B 4 4 i ~ u n i r ;q~&:4~&;?ni-4?~, 7u I I 4 I 4 I

(ab j ~ iu~n; lnnwkdiu~nu) ~dunm.ralur:~q?4~~4rou1n7 u;li!;iana4o;iP mnq-

Study-rrate oacillatian under harmonic driving force v J 8 4 '4 n'?uun'l~ut~ t n n o u n ~ u u m i ! ~ ~ ~ : t fludwuouwcm(nr up rurn0

P(t) * Gcoeot ( n .eb)

I 8 4 4 J A n'?m~no~dllnlfu 1 fun71 steady-etate solution 91dudn4nlr rnnauMao4oynln

Absorptive and elmtic amplitude A v I ' J u n u ~ t rir:'lnn~unotdtll3~nua:nine~ iddo3ulunlroad%i tanmcr :uu

0

I r ld~ulrfia~ulu1~~3~1~urocdo~0'url~~h r us: n ~nun~undquudcnacn~rood+a-

v v I n w n v 1 Ian A un: B ' l m ~ u l i ~ u LLW~I q t ) ncluaun?r (....I niulrnulnl

A UfIS B I;&

nqn4n Aab 11Un-h absorptive amplitude ufii Ael 1 fun;q e l a a t i c amplitude 4 ( u q d e l a s t i c amplitude L ?unitnu r Su dispere ive ~ l p l i t u d e ) nqtn L 17, fun

4 @ J 4 4 4 v r a n i n ~ n tfiuae ~ ~ a q h n r i : ; q nqr pnn&n?h~ loaunqu t 9amr suulnCuiiia&na~n v I v

L ~ l i n n 4 ~ ~ ~ s i ~ t dqun4 A , ~ C O S U ~ ~ ~ n q n n u n n m : ~ ( t ) unoqh 4 4 ' W V 1 t~nunnonnu~rauoatn~road~m am'ateady-state ?:nn?u ~ h q u u n d i u a ~ ~ m p q q u

4 4' v q r m m n'lh4amrlnl ~ ( t ) iiauf4 F ~ C O W ~ a ~ n 7 u n q q ~ l i 7 t ( t ) n?iuti?nmsln~ I 4 a 4 44

d i ? ~ ~ ? q 8 \ r l d n t d f i ~ f . j ~ a : d ? ~ ~ ~ d d t~ldnw

L?Q T f i n n i u n i r ~ ~ ~ j n l m n ? u ~ ~ ~ i ! u a f u

< coaot sinot > - & sin20t > - 0 (n .ad) 4 @ . I f i~uLr iP: I in?i7 ~nsuniu 1 7arnaonMifJroul;anr:uun study-state LIU

J I 1 4 1 v p h d l m i r (m .bc) M steady-state ~ k ~ ~ i ~ d : d u ~ j h n ~ n i n 1 ~ 1 S U R I R J ~ O U ~ J U ~

I t J 9?a LRri:;?n+fivu:~1:1n7ht;uI;rruu ~ ( t ) ; ~ ( t ) niudunir (m. n l ) I u ~ ~ i C u n \ ~ d n

I 4 # I 4 .I 1 J I n r ~ ~ u ~ u n b u ~ : u n i ~ ~ i f i ~ r a i o n u n n a m u ~ r o u r a ~ & n ? 6 j n 4 ' 1 l;air cuuaa:n?~Jn~~l au1hil84pinnain aUnnlu L ~ l r ' 4 d ~ ~ P l ~ i : c l i l ~ t ~ ' 1 1 0 4

J )rfi41udr f iu~l i l fu d ? ~ f i steady-state oscillation ~l f4~ i~ lnnud:du E 8~

4 2 ' 4 l;f~ lnniqnlunu t,, i ;a~fi~qutd~l~hau~~a:n'o~#u' 4 i ; a~&~~~u(m; l~k i l b j ~ q u I v v I 1

~P~oPI: lnqfunql Y q~; u - u ( ~ q w : n f i q q I f i ~ ~ ~ m finfa fa weakly damped ' r" I I ' 4 I I v nqr fuuauq~a'~r : ~ ~ ~ ~ ~ u . ~ a u l n n u l I a s i r ~ ~ ~ ~ u f n u L Q ~ U ~ ~ R ~ I H ~ ~ U I H ? D ~ Y ?~799:18119

Y I I v

~nauqrnm nq u il;quqnidalltutlfu uo nqqulj3w ~ u q a ~ ~ v i d n d t $ i a u i f i ~ h J I U V . I V J a fnu~nt l i anaun i iu~ :u '~~ i~~~ufnud:du~udd?~u~n ~ U H ~ J ~ Y ~fuaq~nq w U ~ ~ U O U L U ~

' 4 4 ' i L u B uo nmul i ~ n a ~ l n a l ; u ~ u a : ~ i . ~ ~ ~ u f n u l a a u u n ~ u ~ n n ; ~

@ I 8

aiiuqn (bonanee) ~ u i u i a ~ h r iuq~wrclqnqr r iduuuie~auqmau 1 gunau~iaer - 4 4 aynqni!aaaa'a I an ificl~;nmudaapur P I ~ ~ ~ ~ L J ~ ~ u u I ~ I ~ ~ : ~ ~ u P u B J ~ ~ ~ ~ o u ~ J ~ ~ ' I I ~ . J

J 4 4 v i ~ a ~ u ~ u w a ~ f u ~ 3 a ~ n ~ ~ a n n ~ t n 1 t u t l \a~n~~unnq~anau: r ~a~u'un~ntaue~:n'71~1n'fi v I 4

.ready-state Inpuqtalhl f 4 % n ? k tnaunqut~a-h;uir:uu9: tau

- 4 4 4 4 . I

1 da po 8a;qna~ P ~ a o l z ldna3urnn * = ro niumnnqnroy P 1 cinna3unn Fnyh5 J J 4 4 nr 4MuP (half-power point n%um auAia4 n n r u P taunt PMuluPrar r;quin

4 J 1 J v nlnmm7nfi?rt4nr PndP lndiiun?nunn,u

v I v 8 t lnd j tnn;qaunqr c m.u~ uu.raanImhao.raunn?r quadratic IU r una:aunqriI

I I V v n?noud t aubu~nua:nqe~Ja~n,u~u d~ruasPn?mu~qu?n'~nn~nounquaunqr (..LC) 1

4 ' , " 4 4 ; ~ t n ~ ~ u n r : t l ~ ~ ~ ~ o ~ ~ n n ~ n ~ n r ~ ~ u ~ ~ i'un;q full-frequency w i d t h a t half-a-

u power i l f a fun 1 Q resonance f u l l w i d t h I ~UUI&I (A@) M. UIO I IUU ~ a ~ h AU maunqr(..w)

I 0 I

T n?uunn?u T - i /r fiUYUn? 7 ~ f ~ ~ u n Y : ~ 3 74 reronanca f u l l r i d t h il+~r~ v 4 8 n ? ~ n n d ~ ~ ~ a ~ n ? u ~ r ~ ~ a ~ ~ ~ n n q t a n n j rnnud~ufun~r o a d ~ a r flnau~aa'd: h

4 I - 0. n~rahmL8n~n?lunno~ ao-1 mode* ro d?u~~nr inac damped free 4. 4 I 4 4

oscillation m?unanumt8nn u1 r~rl:?qn~urngngnrm uo nnnqra3o ul rum

rln; d-ag factor e - (urt lunrQro~nqraas8o i onhunr ~ o ~ l ~ ~ i l ~ n l ~ m r d a a : 4 ' 4 Y ' '4 m?una~umunn nq~nmunno~nirooa+~L a n a t k : Yo innr tnqlrtrt damping

I 4 I amqr ( m . ~ ) d n ~ ~ ~ h ~ ~ r ~ u a u ~ r u . r f i ~ T u n ~ r n n s s . r rnrl:mr r?f:b.j- Y I I I

rnnr~unqnqraiiu~nna.jr :uulna~un??nvfq rnnr ?~qqo)aqnqranaqou?qo'dr: l u .l nr4r;uli r q ~ q u l r n ~ q h ~ x~qnvnnac ionsri~uiunlroafiC ianlnu2null~naqnq1

4 l ariumrno~qnq AU f i i u i7alnlrnna.j i r ~ r i l k m a u n q r (..MI

Frequency dependence of e l a s t i c ~ p l i t u d e

4 Y I r dtiirluidant tliuut.rinoau Focosut l ~ n l r a i u q u b t d 4 ~ n n P j ~ i ~ n ~ r q 1 n n a n d . j ~ ~ 4 H J

elast ic 11;~';9un~11; i n'nnlrOnndurrfi mu icmunw i .raqnaq T :uu un:u'tn?wun d r IL I Y n? lmaBm u - uo . ?in'n5ueuu tmnuululnxuqum?u;~ ~J;m?uh@ '

I I 4 J 1 0 Y 1 , ueauq.jlt nr .j Irueillnn? ~ ~ ~ o a i . j l n t ; ~ ~ n ~ n ' u ~ ~ u ~ n ~ w ~ u e last ic S R ? ~ U H U ~ U I ~ i a2

lit fi~: r lul;;wn 2 2

Fo two - u *el - - 2 2 2 2 (n.b#)

(I: - Y ) +r u

Y l I ; ' ~?n#dd0~nl$ i1?~n71 rr

Transient forced ooci l lat imo

. I v Tnunvn?~unzlnm: 1 runulni x(O) 111: i (0) t TI~IUI~OHI~SIP~~WOJ " 4 1 v dunir d) In94 lfiun?nou%3I!h~ n'n'31nnlf f 2un 'u~nq '~ tq?u steady-state

1 x,(t) ufi:n+mru$21,.1 r l ( t ) PctPfimirrotnlr\ndauvituu h o ~ g e ~ o u

Aabai-t + Aelcoaut + e-(W r t [ ~ l ~ i n u l t '+ ~ ~ e o u ~ t ] (n .n.)

4 ' 4 4 v .I IUO nl LM: B~ I;d l ~ u n i n o n l n ~ n e l i u i t n ~ ilanl~fionnk~~fitnl,; I runuradnlr

. I Y v 4 v . r-%na:nmul j 7 ~ ~ t t ~ ~ i r ~ l i n ? i l n ~ ~ ~ ~ l d ~ i n z l n ~ ~ : 1 rudnunifn?Munlwinm: 1 T unu I v I 1 I 4 4 4 ayninlulngn~un~uuneui.rlr r r u i u n ~ i l n n u o ~ ~ a ~ t - o aynlnrrynutnn?adP

4 " I v ~u~fi;k$u\r i l8on B, - - A = ~ a t l ~ f i n n : ~ i u m \ i u ~ ( o ) - o & l i i f i t d o n nl 1 " ~ n o b m 2 w l i ? i #u;u i (0) ~ ~ l l ~ O Y L ~ U ( J U ~ ~ f 1 l $ ~ f i ~ ~ 1 ~ ~ ~ l : n f r u " w e a k 1 ~ damped

' 4 r l ~ t h ~ r ~ ~ ~ u ~ r o l ; i ( * ) " ~ d i ~ ~ ~ u ~ ~ , * l u r : ~ ~ i ~ ~ u t ~ ~ u ~ ~ ~ n i ~ ~ a ~ t ~ ~ ~ n w n n v ' 4 " v I d t : u ~ ~ ~ n u u h l w f i i u i t o ~ i d n . ~ 1 n ~ ~ ;(o) = U A , ~ + 4 4 4 l ~ O f I 9 l ~ ~ ~ f l 4 1 1 ~ qlRROl.4

I I

l u ~ i r l n a ~ i n o k ~ ~ l d ~ l r a 1 r i ~ i u i r m 8anl; nl - -nab M ?O ;(0) ,. (10 - ul) nab (n .ab)

x( t ) - ( 1 - e - ( Q r t ~ { ~ a b ~ i m t i + ~ ~ ~ c o r t ~ - { I - e -(urt)xs(t) (m.md) 1

1 da 5 ( t ) ~ d u n ? n o u steady-state i ~ c i u L i L ~ I ~ H ~ I U ~ ~ D J L L I c i n k u in'if inqiu J n n i t a a ~ Q m n m i u a r r u m 8 L Y II~FISI~IDULJULLUU study-atat. ~ L L ~ L L Y ~

= Fo/M

fiu",dlJnll ( m .mm)lm; Fo {cosut - cosuOt) x ( t ) I - - ( n . n r )

J ' - 1 n u n i r (m.mr) $ 1 ;IH~UDJL ilumiiinii nuriulnsadna.rniuuirluunM I ,il;~ilnuirinw-u:

4 l 4 8 I I ndq?In';7~ud~nradnqraad+a 1 nnunnauaunoan 1 ?aidt \~t~funiuaaI%oui41uau~n

nrdil . beat. I ~hnrbln;\i iurknrii u - r, ua:- r - o 1nur~lmjdo~n~iI;~lucruufu

v transient beat l n l l d n 4 1 ~ ~ d n .b - c .

!

~ o 1 d n ' ~ i r o l i b ~ d n ' ~ n a a ~ n i ~ a o d ~ n i nn9innir i lnui transient .nap

d 4 4 I W v v a 4 f:u:L~,nT m i u t I,LCIUU~~UOU damping d?~lrnn:n'.rIn ou id l r f i n lu~uo t m a ? n I x au~~~~n i ik lu lnn?1 in2~uL i ,uqn5uc I~u dampingnr mfI;filul,na:n~ I HI;DUMIU

4 4 4 urnln i u o ~ m n ~ damping ~u;4inwl : ( i~ni l inaaun l H j i o u i ~ u ~ l ; n q u ~ u t ~ n ~ 7

v 4 I t7id3qns:oada~ a n f i a t n u n 4(u0 - u~a:ak4clu~uf ioe&a t at 4 4 u'nv 1 d u u i i ~ a . m ~ u n ~ v d i h wo- u i ra i l dup ing 1;l l lhri i44lul~ i l u l d ~ u o ~ ~ ~ n

v

damping i i ~ ? ~ ~ ~ n i ~ j f i o ~ ~ o ~ f i ~ l u L!? 64u"Uil i a l j i l n ~ 4 muiiibamdlnn?b 4 4 0 1 v 1 .

damping $Au;lrlnnIn i uank4 714 t ~ u ~ u u ~ r l ; i 1 1 damping 1u linw: 1 1uiis:~u7q v I 444 J 1

damping n)ll;!nHluld t r lnl?l?Innmn: ~.u~rn?%ol:nubna~ 6tditn~;l t ~ u n r a u r 4

I v a 1 dunlrX: tl;lfidmlr (m.rr) rlt rihcll t - - v

4s

t t ~ ~ ~ i n ~ ~ u n a u , i ~ o n u m i ~ u ~ ~ n a o ~ n ~ ~ ~ a ~ a ' a t n m n ' o nab(ro) do 8

4 I ~nith~u;ioi!u,n n?luqf dfiJaandt t minnr LUI; n(uo) unit MIL n&(uo) I v 1 1 A #

t r d a ~ u l n l n u t ~lsuin'uununit (6.r.)unr (m..b) nquo~ t#u3fu t t 1 ~ 3 ~ n ~ i ~ l n ~ ~ m v

ahmumy n ? n s u ? ? d ~ ~ u J ~ ~ n n a 4 ~ 1 t o o d ~ ~ t ado nel(u) ~ ~ d ~ n d t t lainn; tu V 8 1 I n n(r) ditifiu r n i ~ l n n v m ro $A 1n'iri.u nel(r) a4 t t ~ ~ ~ u ~ t a n ' ~ r ~ l ~ l n u LJ~IIU

t~uu9lndmif ( m . ~ b ) ua:(m.rd)

n.b n~tojiU~nl~truudo~itnloorr~fnou I # 4 4 4 l umd . L rmu?iunft: mode a o ~ t r u ~ ~ d ~ n t (~mo;idddt:ndnntso~?-

I J I I : rmusinn3mu4iloinlr LHi;ouri.uJ auui;~lriinouidiu ua: t rllnorn4 dampiup b q v

I 8

n i t {nu, t w d ~ u t do t twq d m i n g ts~uinnmu~:nu~~una: mode (9:

twibufi 0 m 4 h e ~ ~ i - i osc i l la tor kJ~uuinr mode ~:d$nm:oin i t

oogatan uaril duping colutaut r t~unod(i3iio4 ~?'l;dffn~w: decay ti= v

tgun~~(i3f i to~n3s ~ i r r u u i ~ t ruu~nmmi t tzn damping oiqP:fuljtkuwa 4 v

t ndouklnfiwu4 mode hunaiqq:d dampiup constaut tin: decay v 8 1 I v J

t-e tvhfuln fi t & h a u i r a o ~ t : u ~ s o ~ ~ n g u t w 3 ~ u ~ ~ u ~ ~ u ~ 1 a u ~ ~ 3 4 t u i i uognga a 1 4 '

n*Jdodm14~nn~~Ul nooun tnfiuut;a: o d e $ 4 6 mode )Ijdo~q:jf decay time 1 I I V 4 t d l ~ u ~ ~ u r : u m u y n l t t3n dt.ping i ~ u h i u modes (i1ou14 IN ~ I ~ I ~ ~ P M R ~ U

J ~ri.ugn{d4da4ihnJui4J:~noi n'l~wrurnddf ~gnnnuhdrmcm tn'nJ f r i c t i o l u l 1

.damp* aitlm;ltn'n damp* IQIII;~.IL~~~ fiC mode 2 (go m d e d t d ? ~ ~ n 1

dnifoncl) 1 damping colutant uinn;i mode 1 $uffo r2 >> r iia:n?b I v J

Me 2 ~ t ~ a i n i t r n n ~ & n ~ i mode 1 n w r2

4 4 J " v v ~i l r i l l lauun?iunao4afrLnaouasi4r i l u s : ~ ~ u u ~ n u ~ ~ . r r a t b n t l n i r ~ n 4 4 4 4 n~unk-4iurinu7sl nsouv \du#q;~aern?iunut 4 1 nneu u 1 timu;i 1 rinoriumu~ht:

dampirig constant 11a: L . ~ n i n i r a n a ~ f i u f i n i r ~ o ~ i ~ l i ann;iriidr:aaq mode 1.1

v O~:uuli;niutd ... dunnlrn a ua: b dwn M i d i ~ nat4dl;mudfion

v I

~ u L ~ u l L u u t l ~ 4 ~ tlfi:n?Mun~Yunn:~n{d damping constant r 1.111111 - 4 4 . l V 8 .,z n ~ n i t ~ ~ a ~ n i r L ~ R O U M ~ ? U I T ~ ituu \ndiun4u

w a ' = , % , 11 a - #(Pa - yb). - MI'$, + ~ , c o s ~ t (n-dn)

2 g Mode 1 : $a = $ b to1 = 11 *I - ( m - d g ) Mode 2 : $a --gb 2K

43 ~ U M U UP: q2 1 $ no-1 coordinate^

%in'lua4 1 n'u?b \ 1 IfliuitnM? no-1 coordinates y tin: $2 rqqllunlr (n .m) v v i

an: . ~ ~ n u u ? n n u n i t (n.dn) un: . l a i n ~ u f u 1 r i h

n ~ r i : ~ i ~ u n i t . us: t..u) ~ i t ~ p ~ u u u t h drip.0 4

damped hanwrnic o r i l l a t o r IIll(llMU1: #tlh no-1 coordlnate flT:Wt$! l ~ i l o u f i ~ h n i t a s d ~ n ~ anuuus~;luilna;it;iUllu'u3a x ~in4111110 Y:, damp*

4 -

constant r un:hr 4lnnau\h l o , c o r t un: n o d coordlnate )2 JTmminu a 4 n'luasI&?fihu u?n w n i n ~ n t t l l i ~ mi , damping c - m t r u~:u'utan&ju

I as ' e o c o ~ c n l ta~d& lank toonirluaurmfi f~ik L r i~i~iratiuun?nauiidni~:l~ ltlbuudardqrru un: i2 uunaan,iniTi& ~ r m 1 d s u ~ w mode n r r h f i l n ~ r m l

one-dimensional forced osci l lator 6duuI;n: .ode 9:j absorptln -1i-

tude L L ~ : e las t i c amplitude I JuIo~IIsIJII lad u ~ : i / n ~ i ~ ~ d n h i ~ ~ ~ u ~ C u n ~ i u d

L AIL b ~ T i u n q t ~ ~ t t 1 n u r:uunirand& 1 n m d ~ j i l A 1 f l ; ~ i ~ ~ i t r u i n i r lndnu 4 v 4 v nnoddo4gnqu a u ~ : b l u a ~ ~ i f i a ~ n i t (..ec) Uni(n.6b) 1 t i l l

& v k4Uu absorptive aqlztude nOJ(nqU a 9: 1 ~ ~ ~ 1 1 ~ 1lJlIflJ absorptive a~@It.d. 4 Y v n nuirqnn4tia4 .odes d?u absorptive ampi i t& mqnqu b 9: L.~U(IIIIPI~J

absorptive amplitude ~14df14 lodes M ? U ~ 4 1 $u?~u e las t i c amplitude nn q n 1 * v j u a ~duua~?uno4 e last ic amplitude nlnqin .odes un:d?unaJgnqu a

" 4 4 4 l 4 Lgud?uuan(qaaa4 .odes fida4.2~ LM~?~UOLLIJ l n n a u ~ n ~ ~ u n ~ i u n ~ r d e l i f q mode 1 A 1nm)l1;nqtLnnaunw4 a un: b 1~u~tlnirbnw:aod IIIO~. ~ ~ ( h n f i u n i r o a u ~ a l s n

I

auiqddt:)

2 J n .c uffnuni? r ~no~uin'lulruuuovIIn~ooVMf mou plot iludfiqm qbsorptive bar

e l a s t i c tiu nmud

4 4 A V V 4 study-state nwm qnq~ tqr: l~~5u~nfluf l i mn~?uout,~ i naau i r ~ ' 1 n u n ~ ~ u o mode

I J v i nu i lu~~ur:uuao~i tn3oor~i 'nnu~ i rilnitnrnuaq ua: i r7numn .twy-.tate

@

binlilnnn~u~~:qn~u i Junqr t quf i l ino j;~dulr~nun'n: n a l ~ i Y v~+ir7taq ' 4 1 J

I s u u n l j l u a n ~ ~ : ~ u ~ ~ n u u i ~ ~ n t n ~ u I ~ u t ~ i nmouBinqwo r Iv ~ u t n i r l r s ~ i n i l u n s 4 I v v I tnnzrau i m ua:lun?wun;lilur J i naount :n?opngunt ~ fq lnuu l t nqrnt :n? inuu

I 1 1 4 v l1u4oul.r in'uqraruv'h: mode u(; absorptive amplitude ~ U ~ I I I U L ~ U ~ ~ D J AU )I'J r; uLnrl:;i absorptive amplitude n lu~ IJ~ ; l~ f?n i fq it(; elastic q l i t u d e

Y 1 J J J v 1 Y I Y I ' Y ma~n~nqin~1:nnq7un i d a u u u ~ ~ t ~ i I l p ~ ~ f t l a ~ n l ~ l n l u i l dampin8 ~ P t l u l n t : ~ ~ ~ : ' 4 # * . I ~ h a n l d ihdnl?:~UidaUUU~nP k n f i t : ~ t f i d b n ~ o l : ~ 9 1 ~ \ n 9 1 0 f i 1 ~ 0 ~ 7 ~ d ~ ~ ~ am fiui

J L I l ~ t n t n u u d i ~ ' dampin8 nl~ddn?Wunli F - o LLvu LLR:IIU~U?UH~~ L ~ U . U J B ~ U ~ C ~ W :

4 v Y 1 Y i o lnlr qo4riu iua i n?lnaoriulw (ad i r q

1 II t : U?DIUUUBID t UBP ( continuour approxja~tion) I

1 nitJr : u ~ u ~ u n o r u o ~

d S 4 t r i n w d i )n(t) mi i J~UUUII~P t$n;au L&I u ik;l~Cuiu I I W U ~ ~ ~ U 1 V " 4 ' v v 4

71gnquw'~unmaqlns1u?1 moa~qnqu n ~ ~ i k + u n ; ~ d ~ a u d z Snit i n k u d ~ h -4 4 v

i &I i b7fi0n{u n k f i n i r L naaunoaqnqu u 4 = i i r in taa iu iu l~n '7u

- *(t+a. t ) *(z,t) + a @ ( ~ . t ) + lp2 a2*(z,t) + az . . a z2 @*l ( t ) - *(w. t ) - *(..t) - a = az + 4.2- +..

az2

2 - ...... CPUYU **lo *n a 11. a z + b2%+ - az

2 clL b 2 q + - . a......

*U - *Id a z a t v

unuiinunir ina(irf (rlu#Punu pn(t) - a2~(s .c) /at2) Iunmi t (l.go) ~ t i I n

s I v ' cla..ical equation u i n i u n i i n i ,,,, t h p u

@ 4 " 4 4 . g - 4 4 4 ' XP A ua: B t h n i n ~ n v i 1 n ~ i n ~ n i ~ : ~ c l u t ~n nauninnnuudrnqquulqnavnuuuPu . I 4 4 I fduafuniiuin ua: n;iunoiiuiniiiian~ii~o rod. Uu L ad

J J 4 ' 84 nni& : uupar 4 L nsaun r :x~'rln~u.nd.r . = o d d i ~ x h u ~ n ~ u u n l r gnoanuqnA9n 4 4 ua:ouquldrln z - o 8 4 z - L ~ J M z = L iidd?~JnRnLuCd fldiu;q\tq"l;urr

J v 4 I i L nnaunr:ntldar :uun,u~mmvhn?q cutof t CuUti~n A(.) q:i&anapL mt:u:

fl v I , r ~ l u A ;qr:utliinmuuq?um n k b L ~ ~ q u q n ~ 8 ~ n f i ~ ~ u B u t u t i c a u ~ j u ~ u u

J . I z - L b n a l ~ ~ q a 1 ~uD~u;I ~u~z~nh~ z = (r;~amq; k+"# TII

$(z , t ) - A(Z) coe(wt + $ ) (m .bd) ' 4

n q n ~ n r 1 fan;? amplitude attenuation c o l u h n t ufo 1 h n h 1 P I I I

attenuation constant i f ~ l l ~ ' i n ' ~ ~ ~ f l d 7 ~ amplitude a t t e

n7?uui7aoJ A(=) nbunr $u

n'7U?buroJ r l ~ u ~ ? l l l u l ? 6 L fun;? attenuation length

1 - = 6 = attenuation length K

( n .ea)

' J 4 ~ u i d i f d u i d a u i d ~ u f i m n a i u ~ r : M i l d attenuation constant r f f ~ ~ f i n a u 1 4

(an;~l]Lllill ~ u a U~:P?U?UR~~U ( wave rider k f i ? M f U f ~ S U ~ d ~ # f l r L S U ~ ~ 7 I I I I

d7u attenuation maHu7uf:u:nia k ISUISIU~U\ 1 ~ i i~um~u~ur:u:nw H?UB.I f r

4 I v l # ~ 7 f h attenuation length 6 U L ~ : ~ ~ ? U U I ? ~ O U A ~ l J 7 ~ ~ ~ l ~ n f i 1 ~ f l U # f l 6 L~ IU

v

r:u:nia)l~fu attemation nquurinlna; c1 uar .A \ B u r : u r n i ~ ~ m T u n i r r

i

An - A sinkna + B coekna f i u " ~ An+l = A ein(kna + ka) + B cos (krra + ka)

An-l - A sin(k& - ka) + b cos(kna - ka) *e l + = 2A sinkna coska + 2B coekna coska

- 2 coska(Asinkna + B coskna) - 2 coska An 2 2 2K

0 = 0, +- ( 1 - coska) 0 kl

2 W =

4K 2 & UJ + - s i n

0 H 2

a 8 mode p 1 n fd'i~~ tm:nma:lurl i d r ? ~ c e ~ ihuuu y o r a i a l 2- i;o

i I I I

uon Cb) m?iu#ufluabou I I I 2

0, lrFu 2. ( c ) mmu

# ~ i r ~ & = n i ~ ~ # ~ d t i p n

A(z) rraz z ainTurr7u

~nkunmud w 2hifiu

3 J n .d uamn?iuWu~u<

n7.tn.r=wuvnt;?*n?7ud

(a) m7iudqvn;i high-

frequency cutoff mdu

rhinp r nutuz4n ran

(b) b-rwnwddispersive

rrdu r had&

(c) rr~iudvi9n;i 1-

frequency cutoff:

exponent ail waves.

3.1 Plot a diagram of a damped oscillation whose equation is given in the

-0.lt form x = e s m.

3.2 The equation of damped oscillations is given in the form

X = 5 sir+ m.

Find the velocity of an oscillating point at the moments of time: 0, T,

2T, 3T, and 1T.

3.3 An equacian of undamped oscillations is x = ain2,5+t cm. Find the

displacement from the position of equilibrium, the velocity and the acceleratisth 7

of a point 20 m away from the source of oscillations for a moment of t - 1 s after the oscillations begin. The oscillatios propagate with a velocity

3.4 Verify that the solution

satisfies the equation

when r2/4m2 = ~ / m .

3.5 Show that the bomdary condition x - A cos$ st t - 0 imposed upon the general solution

-rt12m(cpiu't + C2e x - e -bft1

for damped simple harmonic moyion, require&

I c1 l e e A 2 md c2 - Q ;I4

129

3.6 A capacitance C with a charge qo a t t - 'O discharges through a resis-

tance R. Use the voltage equation q/C + I R = 0 t , ~ show that the relaxation t i m e of t h i s process is RC seconds, that is,

(Note that ~ / R C is non-dlmenstibnal.)

3.7 The equation & + Kx = Fo sinut describes the motion of an undamped simple harmonic osc i l la tor driven by a force of frequency w. Show, by eolving

the equation in vector form, that the steady s tace solution i e given by

F* sinot x = -

2 where yo K = - m m Sketch the behaviour+of x versus w and note that the change of s i p as

passes through wo defines a phase change of n radians in the diiplacemcnt.

N o w show that the general solution for the displacement i e given by

Fo sinwt x = + A coswot + B eiaoot

where A and B a r e constant.

3.8 In problem 3.7, i f x = 1; - 0 a t t - 0 show that

and by writing w = wo + AW where AW is small, show that , near resonance,

Sketch t h i s behaviour, noting tha t the second term increases with tisla,

allowing the osci l la t ions of grow (resonance between f r ee and forred oreillation)'.

3.9 An a l t e rna t ing voltage, amplitude V is applied across an LCR s e r i e s 0

c i r c u i t . Show t h a t the voltage a t current resonance across e i t h e r the

inductance o r eondenser i s QVo.

3.10 Show t h a t i n resonant LCR s e r i e s c i r c u i t the maximum p o t e n t i a l across

2 J5 t h e eondenber occurs' a t a frequency w - wo(l - 1/29,) where w z = (LC)-' and Qo - U,L/R. 3.11 See Eq. (3.40) and f i l l i n the a lgebraic s t e p omitted i n obtaining

-t/r t h e . r e s u l t E - E o e ,

3.12 Show by d i r e c t subs t i tu t ion that x l ( t ) a s given by Eq. (3.3) is a

solut ion of the damped harmonic o s c i l l a t o r equation of motion, Eq. (3.2).

3.13 Show t h a t i f xl ( t ) i s a solut ion of Eq. '(3.1) f o r a driving force

Fl(t) , and i f x2( t ) i s the solution for a d i f f e r e n t driving force ~ ~ ( t ) , then the force F( t ) F l ( t ) + Fi(t) gives 'the solut ion x ( t ) = x l ( t ) + x 2 ( t ) ,

I

provided t h a t the i n i t i a l conditions x(0) and x(0) f o r the superposit ion

a r e a l so the corresponding sums of the i n i t i a l condit ions, i .e., provlded

x(0) = xl(0) + xi(0) and ;(o) = i l (0) + Gj0).

3.14 Show by subst i tu t ion tha t Eq. (3.14), (3.15), and (3.16), give a

solut ion t o Eq. (3- 13,

3.15 Verify Eq. (3.21) f o r the power l o s s due t o f r i c t i o n . . Verify tha t

i t is equal t o the input power a s given by Eq.(3.30). 1

3.16 Verify tha t the time-averaged s t o y E f o r steady-state

o s c i l l a t i o n is given by Eq,(3,22),

3.17 Verify t h a t t h e half-power p o i n t s f o r t h e s t eady- s t a t e resonance curve

are given by Eqs. (3.24) and (3.25).

3.18 Shcw t h a t Eq.(3.30) give t h e exact s t eady- s t a t e s o l u t i o n t o t h e dr iven

o s c i l l a t o r equation (3.13), f o r t h e case where the damping cons t an t l' i s

zero.

3.19 Show t h a t i f t h e pendulums of Fig. 3.4 are coupled by s l i n k i e s ,

they have t h e same equations of motion f o r t r ansve r se h o r i z o n t a l o s c i l l a t i o n

a8 they do f o r t he long i tud ina l motion shown.

3.20 Sketch a system of inductances and capaci tance t h a t has equat ions

of motion similar i n form t o Eq. (3.50), and de r ive t h e i r equat ions of motion.

3.21 Assume the ionosphere starts suddenly a t a boundary, a t which t h e

cu to f f f tequency v suddenly inc reases from zero t o 20 M c . Find t h e ampli tude P

a t t e n u a t i o n d i s t ance 6 f o r AM r ad io waves of frequency 1000 Kc .

Ans. About 2.5 meters , independent of frequency, as long as the

frequency is f a r below c u t o f f .

3.22 Using the coupled pendulums as a guide, write down t h e complete

d i spe r s ion r e l a t i o n f o r analogous system of coupled inductance and capaci-

tances. We want t h e d i spe r s ion law i n t h e pass band and i n t h e two cu to f f

reg ions o f frequency. '

3.23 Show t h a t , i f w e use the weak-damping approximation and i f we s t a y

resonably near a resonance, t he abso rp t ive and e l a s t i c ampli tudes can be

w r i t t e n (with a s u i t a b l e choice of u n i t s ) i n t he form

where x - (o - o o ) k r 3-24 Suppose w e have a system with two resonances a t frequencibs w and 1

w2 which make equal contributions t o the e l a s t i c amplitude of sode moving

pa r t . For w far from both o and w we can w r i t e ( i n some u n i t s b r o the r ) 1 2 '

1

Show t h a t , I f w d i f f e r s from wl and w2 by much more than t h e i r d i f ference

u2 - wl, t h e n Ael is ( t o a good approximation) j u s t twice as l a r g e as e i t h e r of the two contributions. That is, show t h a t

I

3-25 C r i t i c a l damping. S t a r t i n g with t h e equation f o r t h e underdamped

f r e e o s c i l l a t i o n s , Eq. (3.7). show tha t f o r c r i t i c a l damping the eo lu t ion

becomes

x,(t) = e - ( ~ r t { x l ( o ) + l;l(o) + Wx1(0) J tl . Show that t h i s saad r e e u l t . i s obtained i f you start ins tead with t h e equapion

f o r overdamped' o s c i ~ l a t i o n s , Eq (3.9) . 3.26 Coupled pendulums. Considet a l i n e a r a r ray bf coupled peadultlms

driven below cutoff a t e = 0 and attached t o a r i g i d wa l l a t z - L, a s ehabn i n Fig.3.5. Show t h a t i f JI(z,t) equals A. co$oC as z = 0 , then $ (z , t )

A(z) coswt, where

Yotice that for L + - thlr becores simply ~ ~ e ~ ' .